Thanks, Bob. I'm afraid I can't clarify. What I say they suggest is practically a direct quote. See here: <https://support.google.com/chromecast/table/6207416?hl=en&ref_topic=6279364.>
And, this clarifies that Chromecast support has posted incorrect information. What they are telling users to do is configure the AirPort to do two things at the same time, when you have a choice of only one or the other. Both at the same time is impossible.
It's my understanding that if a router is run in bridge mode with another router, the capabilities of the combination are essentially those of the other router alone
That is correct in the sense that the router in bridge mode will only be able to reproduce the Internet connection speed that it receives from the "main" router. But.....if you are transferring or copying files from one device to another over your "local" network, the "main" router has nothing to do with the router in bridge mode.
You are aware that Apple's own software.....AirPort Utility.....will tell you to set up the AirPort in Bridge Mode if it detects another router on the network, correct?
For example, my "main" router is an AirPort Extreme set to do DHCP and NAT. The Time Capsule at the other end of the house is setup in the correct Bridge Mode setting. The Time Capsule provides a wireless signal with a speed, on average, of about 600-700 Mbps depending on where the device receiving the signal is located. The Internet connection speed from my provider is 100 Mbps, so the Time Capsule produces a 100 Mbps Internet connection signal.
if I disable the router of the ATT-provided modem-router combination, then I could run the Airport Extreme in DHCP and NAT mode
That is correct in theory, but every ATT modem/router that I've ever seen does not have a true Bridge Mode setting, so it is not possible to do this. Check with ATT support to confirm. DMZ mode still has the ATT modem/router operating normally, as a router.....and you will still have a Double NAT error if you try to connect an AirPort router set for DHCP and NAT to the ATT modem/router set for DMZ.
If you have not yet hooked up the AirPort Extreme, Apple's set up software will automatically configure the AirPort Extreme in the correct Off (Bridge Mode) setting for you.....because it will detect another router on the network that is already doing DHCP and NAT. You do not want two devices on the same network both doing DHCP and NAT.....that will lead to IP address conflicts and a Double NAT error.
Good luck on your installation. Let us know how things are working when you can.
Is it too much to hope that, even if only by happenstance, Apple and Google might come up with a solution?
My opinion as an Apple user for about 30 years, would be that Apple will blame the other guy if the other guy's product does not work well with Apple.